Modern engineering analysis has led to major improvements in product design and innovation. Integrated geometric design tools, Computer Aided Design (CAD), with finite element packages are ubiquitous. Building on this success, researchers and practitioners would like to extend the application of engineering analysis to problems of existing systems, i.e. ones that were not created within a CAD system. Such domains might include forensics, reverse engineering, analysis of the natural environment, and the life sciences. An emerging area of great interest is bringing the success of engineering analysis enjoyed in product design to medicine. The goal is to be able to engineer specific medical treatments for each patient, referred to as patient-specific medicine. The common thread linking these diverse domains is that the geometry of the problem to be analyzed does not come with a ready mathematical description, as in CAD, but rather in the form of a set of discrete points. The challenge, then, is to develop an analysis-suitable description of the problem domain directly from the point set.
One option for developing an analysis-suitable geometric representation is to mesh a point set for use in finite elements, as exemplified in FIG. 1. This option does include some difficulties, in particular, developing a mesh that is topologically sound, i.e. does not intersect or overlap, does not have holes or gaps, etc., can be quite difficult. In practice, an analyst often must manually assist in the meshing process. Such efforts are time-consuming and expensive in human time. Efforts have been made to automate the task of generating quality finite element meshes for complex geometry with some success. For example, generic template meshes have been mapped to patient specific geometry in order to curtail human effort [Salo, Beek, and Whyne (2013), Baghdadi, Steinman, and Ladak (2005) and Bucki, Lobos, and Payan (2010)]. Yet, simulating large deformations proves troublesome for these meshes.
Another option is to use a mesh free method. The computation of mesh free shape functions requires each point to be associated with a compact support and that the compact supports overlap, such that they cover the domain. Each point must also associate with a portion of volume. Many of the proposed methods for computing particle volume and supports rely on Voronoi diagrams or other mesh concepts. Generating such diagrams can be expensive in R3, so avoiding them is crucial in developing an automatic and efficient method. R3 is a symbol for a set of three-dimensional points that can be represented as a vector.
Another method, U.S. Pat. No. 8,854,366 B1 to Simkins et al., teaches building an analysis-suitable geometry from a discrete point set using a method known as the Reproducing Kernel Element Method (RKEM). The RKEM method is based on a mesh of the problem domain. The subject of the disclosure was how to develop a smooth analysis-suitable geometry from an arbitrary point set, but the method still requires one to generate a mesh of the point set, then develop the geometric representation along with a method to help determine what the mesh should be. The Simkins patent necessitates the generation of an initial mesh, referred to as the “surface triangularization,” to generate additional points for the mesh-free representation. The current invention obviates any requirement to use an initial mesh. For the exemplary application discussed below, a voxel mesh is initially used, which is easily obtained from medical images for determining the representative volume associated with each particle (also called a mesh free node or vertex). The distinction is that the present invention does not require generating a triangularization of an arbitrary point set. The exemplary application simply utilizes the voxel mesh that is provided through the imaging. Another distinction between the current disclosure and the issued patent is that the previous method requires that any new points added from the mesh free method, have a pre-image point associated with it in the mesh. The current method does not. In the previous method, the boundary of a geometric object and its interior is defined by the points contained in a mesh that is created by the mesh free method known as the Reproducing Kernel Element Method (RKEM). The current application defines the extent of the body by the solvability of some mathematical equations known as the “consistency conditions” as represented by the “moment matrix.” Using the solvability criteria allows for the determination as to whether points are inside or outside the body without referring to any mesh at all. The present invention is the first to apply and exploit this definition to completely define an analytic geometry from an arbitrary point set.
The present invention provides a new method based on a mesh free Galerkin formulation, in particular the Reproducing Kernel Particle Method (RKPM), to form an analysis-suitable geometry of a discrete point set such as an anatomical structure. The use of the mesh free method known as the Reproducing Kernel Particle Method is incidental to the invention. Other mesh free methods, for example the Element Free Galerkin (EFG) method could be used equally as well.
In the mesh free method, no mesh is required for the analysis, and hence the troubles of meshing are avoided. Some implementations of mesh free methods still use a background mesh for integrating the weak form, but this mesh is not required to meet the more rigorous standards demanded by finite elements, and hence, can be generated fairly easily. The present method determines the particle distribution, particle interactions, support size, and representative volume associated with each particle.
Determining representative volume is simple if one has a mesh. In the case without a mesh, the trouble is that one does not know how to account for the overlap of support functions. In the present method, the over-all volume does not affect the solvability of the moment matrix, just the relative volume. This is only a problem near the boundary, where a method estimates the fraction of contribution of the window function volume to be used. Then, once the geometry is defined, the total volume is computed from the analysis-suitable representation and is used to scale all the relative volumes initially found.
An exemplary application of the present invention involves learning the mechanics of the pelvic floor. The female pelvic floor muscles form a complex structure responsible for supporting the pelvic organs such as rectum, vagina, bladder, and uterus. These muscles and associated connective tissues are ultimately anchored to the bony pelvic scaffold, and play a major additional role in maintaining continence of urine and bowel contents. Additionally, these muscles and associated connective tissues allow for important physiologic activities like urination, defecation, menstrual flow, and biomechanical processes like childbirth and sexual intercourse.
The major components of the pelvic floor are highlighted in FIG. 2(a) and FIG. 2(b). A thorough description of the functional anatomy of the female pelvic floor can be found in [Ashton-Miller and DeLancey (2007)], [Hoyte and Damaser (2007)] and [Herschorn (2004)]. The levator ani muscle group, as shown in FIG. 2(b) is the ultimate supporting structure responsible for holding organs in the body, and is rightfully the main focus of most studies regarding pelvic floor dysfunction [Hoyte, Schierlitz, Zou, Flesh, and Fielding (2001)], [Venugopala Rao, Rubod, Brieu, Bhatnagar, and Cosson (2010)], [Lien, Mooney, DeLancey, and Ashton-Miller (2004)]. The vagina, shown in FIG. 2(c), is suspended across the pelvis and is anchored in large part to the levator ani, and less so to the Obturator internus muscles. The vagina, in turn, supports the bladder and works with the levator ani to keep the rectum in a position appropriate for fecal continence at rest and bowel evacuation when appropriate.
Many finite element studies have been produced of the levator ani muscles ranging from shell element models to full volume element models [Hoyte, Damaser, Warfield, Chukkapalli, Majumdar, Choi, Trivedi, and Krysl (2008)], [Martins, Pato, Pires, Jorge, Parente, and Mascarenhas (2007)], [Noritomi, da Silva, Dellai, Fiorentino, Giorleo, and Ceretti (2013)]. These models have been valuable tools in learning about the mechanics of the pelvic floor, but many of the models focus solely on the levator ani, without considering the interaction between the other pelvic structures.
To help address this deficit, focus is placed on the vagina for the examples herein. Since a mesh is required for finite element analysis, the geometry is often idealized, both due to the piece-wise polyhedral approximation and the need to adjust the points to create a quality mesh. The goal was to accurately represent the geometry and produced studies involving multi-component interactions. In [Doblaré, Cueto, Calvo, Martínez, Garcia, and Cegoñino (2005)], the prospects of performing mesh-free analysis for biomechanics problems are discussed with several Galerkin methods being described. They chose to use the natural element method, which is closely tied to Voronoi diagrams, on the basis that essential boundary conditions are easily applied. The ease of applying the essential boundary conditions comes at the cost of generating a mesh of the domain, though the authors say the mesh generation requires no interaction with the user. They showed successful examples involving mostly bony tissue and some cartilage. Mesh-free methods have been used successfully in other areas of biomechanics including the heart [Liu and Shi (2003)], and the brain [Horton, Wittek, Joldes, and Miller (2010)], [Miller, Horton, Joldes, and Wittek (2012)]. In a similar vein, a hybrid mesh-free-mesh-based method based on the Reproducing Kernel Element Method (RKEM) was undertaken in [Simkins, Jr., Kumar, Collier, and Whitenack (2007) and Jr., Collier, Juha, and Whitenack (2008)]. RKEM details can be found in [Liu, Han, Lu, Li, and Cao (2004). Li, Lu, Han, Liu, and Simkins, Jr. (2004), Lu, Li, Simkins, Jr., Liu, and Cao (2004) and Simkins, Jr., Li, Lu, and Liu (2004)].
Accordingly, what is needed is an efficient mesh-free method for producing an analysis-suitable geometry from a discrete point set. The automated analysis capability is demonstrated in modeling vaginal contracture, as might occur in cases of women treated with radiation for cervical cancer. However, in view of the art considered as a whole at the time the present invention was made, it was not obvious to those of ordinary skill in the field of this invention how the shortcomings of the prior art could be overcome.
All referenced publications are incorporated herein by reference in their entirety. Furthermore, where a definition or use of a term in a reference, which is incorporated by reference herein, is inconsistent or contrary to the definition of that term provided herein, the definition of that term provided herein applies and the definition of that term in the reference does not apply.
While certain aspects of conventional technologies have been discussed to facilitate disclosure of the invention, Applicants in no way disclaim these technical aspects, and it is contemplated that the claimed invention may encompass one or more of the conventional technical aspects discussed herein.
The present invention may address one or more of the problems and deficiencies of the prior art discussed above. However, it is contemplated that the invention may prove useful in addressing other problems and deficiencies in a number of technical areas. Therefore, the claimed invention should not necessarily be construed as limited to addressing any of the particular problems or deficiencies discussed herein.
In this specification, where a document, act or item of knowledge is referred to or discussed, this reference or discussion is not an admission that the document, act or item of knowledge or any combination thereof was at the priority date, publicly available, known to the public, part of common general knowledge, or otherwise constitutes prior art under the applicable statutory provisions; or is known to be relevant to an attempt to solve any problem with which this specification is concerned.